The Distinguished Reproducing Kernel

The Distinguished Reproducing Kernel (DRK) is an unnormalized kernel K : Σ×Σ → ℂ satisfying a single self-consistency condition: K is the reproducing kernel of the space of functions defined by the geometry K itself induces on the configuration space Σ. The DRK carries two naturally associated objects. The first is the metric g^K that K generates via its diagonal, which the fixed-point condition acts on. The second is the normalized kernel Φ̂ = K/√K(·,·)², which satisfies Φ̂(A,A) = 1 and carries the contrast function and curvature geometry. The relation g^Φ̂ = ½g^K is a derived identity, not a definition. The point of the exercise is this: geometry need not be postulated. It can emerge as the unique residue of a kernel that is consistent with itself.